On Sun, May 14, 2017 at 5:18 PM, Fred Sawyer <fwsaw...@gmail.com> wrote:
> Michael, > > See the attached slide from my talk. All the various dials work with a > string of this length. They vary simply in where the suspension point is > placed. The pros and cons of the various suspension points were part of my > presentation. > What were some of the alternatives, and some of their relative advantages? Michael Ossipoff > > > Fred Sawyer > > > On Sun, May 14, 2017 at 4:40 PM, Michael Ossipoff <email9648...@gmail.com> > wrote: > >> When I said that there isn't an obvious way to measure to make the >> plumb-line length equal to sec lat sec dec, I meant that there' s no >> obvious way to achieve that *with one measurement*. >> >> I was looking for a way to do it with one measurement, because that's how >> the use-instructions say to do it. >> >> In fact, not only is it evidently done with one measurement, but that one >> measurement has the upper end of the plumb-line already fixed to the point >> from which it's going to be used, at the intersection of the appropriate >> latitude-line and declination-line. >> >> That's fortuitous, that it can be done like that, with one measurement, >> and using only one positioning of the top end of the plumb-line. >> >> But of course it's easier, (to find) and there's an obviously and >> naturally-motivated way to do it, with *two* measurements, before fixing >> the top-end of the plumb-line at the point where it will be used. >> >> The line from that right-edge point (from which the first horizontal is >> drawn) to the point where the appropriate latitude-line intersects the >> vertical has a length of sec lat. >> >> So, before fixing the top end of the plumb-line where it will be used >> from, at the intersection of the appropriate lat and dec lines, just place >> the top end of the plumb line at one end of that line mentioned in the >> paragraph before this one, and slide the bead to the other end of that >> line. ...to get a length of thread equal to sec lat. >> >> Then, have a set of declination marks at the right edge, just like the >> ones that are actually on a Regiomontanus dial, except that the lines from >> the intersection of the first horizontal and vertical lines, to the >> declination (date) marks at the right-margins are shown. >> >> Oh, but have that system of lines drawn a bit larger, so that the origin >> of the declination-lines to the right margin is a bit farther to the left >> from the intersection of the first horizontal and the first vertical. >> ...but still on a leftward extension of the first horizontal. >> >> That's so that there will be room for the 2nd measurement, the >> measurement that follows. >> >> And have closely spaced vertical lines through those diagonal >> declination-lines to the right margin. >> >> So now lay the thread-length that you've measured above, along the first >> horizontal, with one end at the origin of the declination-lines to the >> margin. >> Note how far the thread reaches, among the closely-spaced vertical lines >> through those margin declination-lines. >> >> Now measure, from the origin of the margin declination-lines along the >> appropriate margin declination-line, to that one of the closely-spaced >> vertical lines that the thread reached in the previous paragraph. >> >> With the left end of the thread at the origin of the margin >> declination-lines, slide the bead along the thread to that vertical line. >> >> That will give a thread length, from end to bead, of sec lat sec dec. >> >> ...achieved in the easy (to find) way, by two measurements, before fixing >> the thread (plumb-line) end to the point from which it will be used. >> >> I wanted to mention that way of achieving that end-to-bead thread-length, >> to show that it can be easily done, and doesn't depend on the fortuitous >> way that's possible and used by the actual Regiomontanus dial, whereby only >> one thread-length measurement is needed, and the only positioning of the >> thread-end is at the point from which it will be used. >> >> Having said that, I suppose it would be natural for someone to look for >> a fortuitous way that has the advantages mentioned in the paragraph before >> this one. >> >> And I suppose it would be natural to start the trial-and-error search >> from the thread-end position where the thread will eventually be used, to >> have the advantage of only one thread-end positioning. >> >> One would write formulas for the distance of that point to various other >> points, with those distances expressed in terms of sec lat and sec dec >> (because sec lat sec dec is the sought thread-length). >> >> And I suppose it would be natural to start that trial-and-error search by >> calculating the distance from there to the right-margin end of the first >> horizontal, and points on the right margin...because that's still an empty >> part of the dial card. >> >> And, if you started with that, you'd find the fortuitous method that the >> actual Regiomontanus dial uses, to achieve the desired end-to-bead >> thread-length. >> >> (But, if that didn't do it, of course you might next try other distances. >> And if you didn't find a one-measurement way to do it (and can't say that >> you'd expect to), then of course you could just use the naturally and >> obviously motivated 2-measurement method that I described above). >> >> The distance calculations needed, to look for that fortuitous, >> easier-to-do (but not to find) one-measurement method are relatively big >> calculations with longer equations with more terms. >> >> ---------------------- >> >> By the way, I earlier mentioned that I'd verified for myself, by analytic >> geometry, that the Regiomontanus dial agrees with the formula that relates >> time, altitude, declination and latitude. That involved big (maybe >> page-filling, it seems to me) equations with lots of terms. When a >> proposition is proved in that way, that proof shows that the proposition is >> true, but it doesn't satisfyingly show why it's true, what makes it true. >> >> The naturally and obviously motivated construction that I've described >> here is much better in that regard. >> >> The only part that gets elaborately-calculated is the finding of that >> fortuitous, easy to do (but not easy to find) way to get the right >> thread-length with only one measurement, when the thread-end is already >> positioned for use. >> >> But, as I mentioned, the desired end-to-bead thread-length can be easily >> achieved by the obviously and naturally-motivated two-measurement method >> that I described above. >> >> Michael Ossipoff >> >> >> >> >> >> >> >> On Sat, May 13, 2017 at 9:23 PM, Michael Ossipoff <email9648...@gmail.com >> > wrote: >> >>> When I said that the vertical hour-lines should be drawn at distance, to >>> the left, from the middle vertical line, that is proportional to the cosine >>> of the hour-angle... >>> >>> I should say *equal to* the cosine of the hour-angle, instead of >>> proportional to it. >>> >>> ...where the length of the first horizontal line, from the right edge to >>> the point where the vertical line is drawn, is one unit. >>> >>> Michael Ossipoff >>> >>> >>> On Sat, May 13, 2017 at 7:03 PM, Michael Ossipoff < >>> email9648...@gmail.com> wrote: >>> >>>> Fred-- >>>> >>>> Thanks for your answer. I'll look for Fuller's article. >>>> >>>> One or twice, I verified for myself, by analytic geometry, that the >>>> Universal Capuchin Dial agrees with the formula that relates altitude, >>>> time, declination and latitude. >>>> >>>> But that wasn't satisfying. Verifying a construction isn't the same as >>>> finding one. Without knowing in advance what the construction and use >>>> instructions are, I don't know of a way to design such a dial. >>>> >>>> ...or how the medieval astronomers and dialists arrived at it. >>>> >>>> But there's an exasperatingly tantalizing approach that gets partway. >>>> ...based on the formula for time in terms of altitude, latitude and >>>> declination: >>>> >>>> cos h = (sin alt - sin lat sin dec)/(cos dec cos lat) >>>> >>>> Dividing each term of the numerator by the denominator: >>>> >>>> cos h = sin alt/(cos dec cos lat) - tan lat tan dec >>>> >>>> If, in the drawing of the dial, the sun is toward the right, and you >>>> tip the device upward on the right side to point it at the sun, then the >>>> plum-line swings to the left, and the distance that the plum-bob moves to >>>> the left is the length of the thread (L) times sin alt. >>>> >>>> So that seems to account for the sin alt, at least tentatively. >>>> >>>> Constructing the dial, if you draw a horizontal line in from a point >>>> on the right-hand, side a distance L equal to the length of that thread, >>>> then draw a vertical line there, and then, from that side-point, draw lines >>>> angled upward by various amounts of latitude, then each line will meet the >>>> vertical line a distance of L tan lat, up from the first (horizonal) line. >>>> >>>> So the distance from the horizontal line, up the vertical line to a >>>> particular latitude-mark is L tan lat. >>>> >>>> At each latitude-mark, make a horizontal line. >>>> >>>> From the bottom of that vertical line, where it meets the horizontal >>>> line, draw lines angled to the right from the vertical line by various >>>> amounts of declination. Draw them up through all the horizontal lines. >>>> >>>> Because a latitude-line is L tan lat above the original bottom >>>> horizontal line, then the distance to the right of the vertical line, at >>>> which one of the declination-lines meets that latitude-line is L tan lat >>>> tan dec. >>>> >>>> That's where we fix the upper end of the plumb-line. Then, when we tip >>>> the instrument up on the right, to point at the sun, and the plumb-bob >>>> swings, its distance to the left of the middle will be: sin alt - tan lat >>>> tan dec. >>>> >>>> That's starting to look like the formula. >>>> >>>> Maybe it would be simpler to just say that L is equal to 1. >>>> >>>> But we want sin alt/(cos lat cos dec). >>>> >>>> The instructions for using the Universal Capuchin dial talk about >>>> adjusting the distance of the bead from the top of the string before using >>>> the dial, and that's got to be how you change sin alt to sin alt/(cos lat >>>> cos dec). >>>> >>>> I guess I could study how that's done, by reading the construction and >>>> use instructions again. >>>> >>>> I guess you'd want to make the plumb-line's length equal to sec lat sec >>>> dec instead of 1. ...and there must be some way to achieve that by >>>> adjusting the bead by some constructed figure, as described in the >>>> use-instructions. >>>> >>>> But it isn't obvious to me how that would be done--especially if that >>>> bead-adjustment is to be done after fixing the top-end of the plumb-line in >>>> position. >>>> >>>> Maybe it would be easier if the bead-adjustment is done before fixing >>>> the top end of the plumb-line, so that you know where you'll be measuring >>>> from. I don't know. >>>> >>>> And then there's the matter of cos h. >>>> >>>> Just looking at afternoon... >>>> >>>> Because positive h is measured to the right from the >>>> meridian--afternoon---and because, the later the afternoon hour, the lower >>>> the sun is--then, in the afternoon, it seems to make sense for a larger >>>> bead-swing to the left to represent an earlier hour...an hour angle with a >>>> larger cosine. >>>> >>>> I guess, for afternoon, the vertical hour lines are positioned to the >>>> left of middle by distance proportional to the cosine of the hour-angle. >>>> >>>> ------------- >>>> >>>> So, this isn't an explanation, but just a possible suggestion of the >>>> start of an explanation. >>>> >>>> Maybe it can become an explanation. >>>> >>>> But I still have no idea how an orthographic projection leads to the >>>> construction of the Universal Capuchin dial. >>>> >>>> (If a Capuchin dial isn't universal, it loses a big advantage over the >>>> Shepard's dial, or the related Roman Flat altitude dial.) >>>> >>>> Michael Ossipoff >>>> >>>> >>>> >>>> >>>> >>>> >>>> On Sat, May 13, 2017 at 3:47 PM, Fred Sawyer <fwsaw...@gmail.com> >>>> wrote: >>>> >>>>> Take a look at A.W. Fuller's article Universal Rectilinear Dials in >>>>> the 1957 Mathematical Gazette. He says: >>>>> >>>>> "I have repeatedly tried to evolve an explanation of some way in which >>>>> dials of this kind may have been invented. Only recently have I been >>>>> satisfied with my results." >>>>> >>>>> The rest of the article is dedicated to developing his idea. >>>>> >>>>> Note that it's only speculation - he can't point to any actual >>>>> historical proof. That's the problem with this whole endeavor; there is >>>>> no >>>>> known early proof for this form of dial - either in universal or specific >>>>> form. (It seems that the universal form probably came first.) >>>>> >>>>> It was published in 1474 by Regiomontanus without proof. He does not >>>>> claim it as his own invention and in fact refers to an earlier >>>>> unidentified >>>>> writer. There has been speculation that he got it from Islamic scholars - >>>>> but nothing has been found in Islamic research that would qualify as a >>>>> precursor. The dial is somewhat similar to the navicula that may have >>>>> originated in England - but that dial is only an approximation to correct >>>>> time. >>>>> >>>>> In discussing this history, Delambre says: >>>>> >>>>> "All the authors who have spoken of the universal analemma, such as >>>>> Munster, Oronce Fine, several others and even Clavius, who demonstrates >>>>> all >>>>> at great length, contented themselves with giving the description of it >>>>> without descending, as Ozanam says, to the level of demonstration." >>>>> >>>>> "At this one need not be surprised, seeing that it rests on very >>>>> hidden principles of a very profound theory, such that it seems that it >>>>> was >>>>> reserved to [Claude Dechalles] to be able to penetrate the obscurity." >>>>> >>>>> So Dechalles gave what was evidently the first proof in 1674 - 200 >>>>> years after Regiomontanus' publication. But as Delambre further notes: >>>>> >>>>> Dechalles’ proof … is long, painful and indirect, … without shedding >>>>> the least light on the way by which one could be led to [the dial’s] >>>>> origin. >>>>> >>>>> So - pick whichever proof makes sense for you. >>>>> >>>>> Fred Sawyer >>>>> >>>>> >>>>> >>>>> >>>> >>> >> >
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